Electrostatic effects in coupled quantum dot-point contact-single electrontransistor devices

S. Pelling,1 E. Otto,2 S. Spasov,1 S. Kubatkin,2 R. Shaikhaidarov,1 K. Ueda,3 S. Komiyama,3

and V. N. Antonov1,a)

1Physics Department, Royal Holloway University of London, Egham, Surrey TW20 0EX, United Kingdom2Department of Microtechnology and Nanoscience (MC2), Chalmers University of Technology,S-41296 Goteborg, Sweden3Department of Basic Science, University of Tokyo, Komaba 3-8-1, Meguro-ku, Tokyo 153-8902, Japan

(Received 6 February 2012; accepted 7 June 2012; published online 12 July 2012)

We study the operation of a system where quantum dot (QD) and point contact (PC) defined in a

two-dimensional electron gas of a high-mobility GaAs/AlGaAs heterostructure are capacitively

coupled to each other and to metallic single electron transistor (SET). The charge state of the

quantum dot can be probed by the point contact or single electron transistor. These can be used for

sensitive detection of terahertz radiation. In this work, we explore an electrostatic model of the

system. From the model, we determine the sensitivity of the point contact and the single electron

transistor to the charge excitation of the quantum dot. Nearly periodic oscillations of the point

contact conductance are observed in the vicinity of pinch-off voltage. They can be attributed to

Coulomb blockade effect in a quasi-1D channel because of unintentional formation of small

quantum dot. The latter can be a result of fluctuations in GaAs quantum well thickness. VC 2012American Institute of Physics. [http://dx.doi.org/10.1063/1.4736419]

I. INTRODUCTION

Recently, quantum dots (QD), point contacts (PC), and

single electron transistors (SET) have been intensively

exploited in applications to single-electron transport, quanti-

zation of conductance, and quantum computing.1–4 The devi-

ces have well defined quantum states with a typical energy

scale of a few meV, which can be exploited for quantum in-

formation processing.5 Also all devices, QD, SET, and PC,

have a high charge sensitivity, better than 10�4 e/Hz1/2 at

low temperatures. This opens the possibility to use them as

electrometers, which are able to detect the motion of individ-

ual electrons.6,7 We studied coupled system of QD, PC, and

metallic SETs in application to terahertz detection. The SET

and the PC probe the charge state of the QD, which is varied

by the absorption of the terahertz photons. A single photon

counting with the SET-QD detector has been achieved ear-

lier.8 The PC-QD detector is less sensitive, but it has the

advantage of fewer technological demands in fabrication,

and a higher operation temperature.9 We combine two devi-

ces together, SET-QD and PC-QD, in order to study electro-

static effects of the devices and compare their performance,

see Fig. 1. An interesting phenomenon observed in PC-QD is

nearly periodic oscillations of the source-drain conductance

of the PC in the vicinity of pinch-off. The oscillations boost

the sensitivity of the PC-QD terahertz detector because of

the large trans-conductance dG/dVg. At the same time, the

effect makes the photo sensitive operation to be intermittent

and strongly dependent on the operation point of Vg. We

show that the oscillations occur because of the formation of

a small dot (SD) in the PC channel, which operates in the re-

gime of Coulomb blockade (CB) of electron tunnelling.

The samples consist of a mesa patterned GaAs/AlGaAs

hetero-structure with two metal gates, PC gate and QD gate,

and SET at the top of QD, see Fig. 1. The hetero-structure has

been grown by molecular beam epitaxy. The layer sequence is

0.4 lm super-lattice GaAs/Ga0.3Al0.7As buffer, 20 nm GaAs

well, 20 nm Ga0.7Al0.3 As barrier layer, 60 nm Ga0.7Al0.3As

doping layer with a Si concentration of 1� 1018 cm�3, and

10 nm undoped GaAs cap layer. The two-dimensional electron

gas (2DEG) is formed 90 nm below surface in GaAs well. It

has carrier concentration n¼ 3.7� 1011 cm�2 and mobility

l¼ 1.2� 105 cm2=Vs at T¼ 4.2 K. The QD is formed in

2DEG by negatively biased QD gate. Aluminium SET is

fabricated above the QD by two-angle deposition-oxidation-

deposition technique.10

II. OPERATION OF QUANTUM DOT-POINT CONTACTDEVICE

One can probe charge state of the QD either with the PC

or the SET. We start with the data taken by the PC probe.

We negatively bias the gates and measure the conductance

G¼ ISD/VSD of mesa channel, see Fig. 2(a). The intensity

plot is a compilation of individual curves G (VQD) measured

at constant voltage applied to the PC gate, VPC, while sweep-

ing the voltage, VQD, applied to the QD gate. The channel is

pinched-off when a large negative bias is applied to both

gates. There is asymmetry in the pinch-off boundary due to

the difference in the PC and QD gate’s geometry. Note also

that the pinch-off boundary is not a straight line. The slopes

of the pinch-off boundary are identical in regions a and c,

while it is larger in region b, indicating higher sensitivity of

the PC conductance to the QD gate voltage, see Fig. 2(a).

a)Author to whom correspondence should be addressed. Electronic mail:

v.antonov@rhul.ac.uk.

0021-8979/2012/112(1)/014322/5/$30.00 VC 2012 American Institute of Physics112, 014322-1

JOURNAL OF APPLIED PHYSICS 112, 014322 (2012)

The change of the slope can be expressed in terms of the

change of the ratio, CPCPCG=CPC

QDG, from 2.1 in regions a and cto 0.78 in region b, where CPC

QDG and CPCPCG are capacitances

of the PC to QD gate and to PC gate, respectively. In region

b, the conductance channel is formed exactly between the

PC and QD gates. This is the region where system can be

used as terahertz detector.9 The QD is gradually isolated

from reservoirs when the QD gate is negatively biased. This

increases the effective capacitance CPCQDG, because of addi-

tional parallel capacitance between the QD gate and the PC

through the isolated QD. The white dashed line in Fig. 2(a)

marks the boundary where the QD is isolated from the reser-

voirs. The line is taken from experiments with the SET,

which is discussed below. In region a, the conductance chan-

nel is shifted in direction underneath the QD gate, so that the

QD is formed simultaneously with a pinch-off of the conduc-

tivity. In region c, the conductance channel is pushed under-

neath the PC gate. The QD is already isolated from the

reservoirs, because VQD is beyond the dashed white line. An

additional capacitance of the QD gate to the conductance

channel, because of the isolated QD, is small since the QD

and conductance channel are now spatially separated from

each other. Therefore, one would expect that the slopes of

the pinch-off boundary in regions a and c are almost

identical.

Periodic oscillations of the PC conductance are observed

close to the pinch-off boundary in region b, see Fig. 2(b).

They were present in more than a half of 14 devices made of

the same GaAs/AlGaAs wafer. The oscillations were highly

reproducible for the particular sample. The other samples

had a smooth pinch-off curve. In a sample presented here,

the oscillation amplitude is largest at VPC��1.3 V and

VQD��1.47 V. They weaken and disappear deep in regions

a and c. One can see from Fig. 2(b) that periodicity is not

related to the conductance quantization of PC which was a

subject of study in a number of works,11–13 as the position

and number of oscillations are not correlated with the con-

ductance quantum plateaus at multiples or rational numbers

of e2/h. There are more than ten periods of oscillations when

G is below e2/h. In order to rule out possible quantum inter-

ference effects of the scattered electron waves, we have con-

firmed that oscillations are not sensitive to the magnetic field

up to 1 T. Observations of similar effect, periodic and

FIG. 1. (a) Scanning electron microscope image of the THz quantum dot

sensor, consisting of the QD coupled to SET and PC. Dotted lines mark the

QD gate, the SET, and the PC gate. The SET and PC senses charge excita-

tions of the QD upon photon absorption, (b) electrostatic circuit model of

the device.

FIG. 2. (a) Intensity plot of G in coordinates of VPC and VQD close to pinch-

off region. There are three distinct regions of PC operation marked by a, b,

and c. In regions a and c, the pinch-off boundary is a straight line with the

identical slopes, in region b the slope is larger. The white dashed line in the

plot indicates the boundary separating isolated (left) and strongly coupled to

reservoirs (right) QD. This boundary is taken from analysis of the SET oper-

ation. (b) Periodic oscillations of conductance G are observed when crossing

the pinch-off boundary. The PC gate voltage was fixed to �1.3 V, �1.5 V,

and �1.7 V. The oscillations are strongest in region b and weaken in regions

a and c.

014322-2 Pelling et al. J. Appl. Phys. 112, 014322 (2012)

aperiodic oscillations, in the conductance of narrow channels

in 2DEG in GaAs/AlGaAs and Si quasi-1D systems have

been reported before.14–16 The nearly periodic oscillations in

conductance of Si quasi-1D channels were explained by the

charge density wave,14 while a similar effect in an experi-

ment with intentional channel doping by phosphorus atoms

was attributed to the Coulomb blockade of electrons tunnel-

ling between quantum dots formed around phosphorus

atoms.15 In experiments with GaAs/AlGaAs heterostruc-

tures, the periodic oscillations were explained by the inter-

ference of scattered electron waves in quasi-1D channels.16

We present arguments in favour of the Coulomb block-

ade origin of the oscillation effect. Stability diagram of the

second derivative of source-drain current �d2ISD=dV2SD vs

both VSD and VPC at fixed VQD��1.5 V is shown in Fig. 3.

The map does not have a monotonous slope, so that the fea-

tures of the pinch- off region are clearly seen. There is a

diamond-shape structure marked by dashed white lines in

Fig. 3. It can be explained either by the Coulomb blockade,

or trans-conductance of the point contact in a regime of

bound states.11,12 In a latter case, the diamonds reflect quan-

tization of conductance of the PC: differential conductance

has peaks when transition between different conductance

plateaus occurs. In our samples, oscillations are not corre-

lated with integer plateaus or, recently discussed, 0.5, 0.7,

and 0.9 ones of e2/h in the G, see Fig. 2(b).

We believe that a SD of electrons is formed inside the

PC conductance channel, which is in a regime of Coulomb

blockade. From the horizontal diagonal of the diamonds, we

estimate the charging energy of the dot, EC� 3 meV. This

corresponds to diameter, d¼ e2/[4e0(erþ 1)EC]� 100 nm, if

one models SD as a circular disk inside the conductance

channel, er¼ 13 is the relative permittivity of GaAs, e0 is the

permittivity of free space. Such a dot can be accommodated

in a channel between the QD and the PC gates, which has a

width of the same order �200 nm. We get, however, a much

smaller size of the dot from the analysis of conductance

oscillations: in a wide range of VQD the period of oscillations

DVQD� 14 mV, which corresponds to a capacitance between

the QD gate and the dot CSDQDG� 1.1� 10�17 F. If we solve

the Laplace equation for a system QD gate—SD with the

intention to get this capacitance—then the small dot should

have the diameter of only �35 nm.17 The estimations imply

that such a dot would contain from 3 to 30 electrons. We

modelled the potential profile created by the QD and PC

gates. It has a saddle shape, without any local minima. In

Ref. 15, phosphorus impurity atoms were intentionally em-

bedded in the quasi-1D channel. A small dots consisting of a

few electrons were formed around impurity atoms, which

resulted in CB peaks in the conductance. The period, how-

ever, was irregular compared to our experiments. Usually

aperiodic oscillations are observed when two or more

coupled dots are formed in the conductance channel.18 In all

of our samples, the oscillations were nearly periodic; more-

over, we do not expect impurities in the channel because of

modulation doping of our hetero-structure. We suggest a fol-

lowing explanation of the SD origin. It is known that mono-

layer roughness of quantum well interfaces gives rise to

modulation of the bound state energies.19 The latter can be

as large as �0.5 meV in our hetero-structure. A lateral scale

of this roughness can be as large as 10 nm, depending on pa-

rameters of growth and substrate misorientation. The SD in

the PC channel can be randomly formed because of this

roughness, which would give rise to periodic oscillations of

conductance.

III. OPERATION OF QUANTUM DOT-SINGLEELECTRON TRANSISTOR DEVICE

Additional information about QD-SET and QD-PC devi-

ces is acquired from experiments with the SET. Particularly,

the SET enables to identify region where the QD is formed.

In experiment, we have applied constant voltage of 1 mV

between source and drain of SET, and probed the current.

When VPC is fixed the SET current oscillates with VQD, see

Fig. 4(a). The period of oscillations has a sharp transition

from 81 to 5.5 mV at VQD��1.2 V due to building up an

extra capacitance between the QD gate and the SET through

the newly formed QD, see Eq. (1) below. We measure the

map of CB oscillations by sweeping VQD from �1.0 to

�1.4 V at fixed VPC, see Fig. 4(b). There are two regions

with different oscillation periods of SET at the map. The

boundary marks formation of the QD. The QD-SET device

can be used as terahertz detector in the vicinity of the bound-

ary.8 We translate this boundary to Fig. 2(a) as a white

dashed line in order to show the region where the QD is

formed. The white dashed line in Fig. 4(b) is a pinch-off

boundary of the PC, taken from Fig. 2(a). The PC channel is

pinched-off below the line in Fig. 4(b). One can expect effect

of charging of the SD in the vicinity of the dashed line of

Fig. 4(b). The effect is determined by the capacitance CSETSD

between the SET and the SD. We estimate CSETSD � 1� 10�17

F by solving the Laplace equation. The effect of charging of

the small dot would be seen as a shift of the SET’s CB peak

position by �0.7 mV. Observation of this shift is unfortu-

nately beyond the experimental accuracy, because the period

of charging of the SD is large compared to the period of CB

of the SET, 14 mV vs 5.5 mV. A small shift of the CB peaks

FIG. 3. Intensity plot of �d2ISD=dV2SD close to pinch-off region. The white

dashed lines are guide to the eye depicting the diamond-shape of stability

diagram.

014322-3 Pelling et al. J. Appl. Phys. 112, 014322 (2012)

of the SET over a few periods is hampered by the arbitrary

fluctuation of the CB position.

In order to complete analysis, we present capacitances

of the electrostatic model, see Table I. In the table, symbol Rdenotes the total capacitance of a corresponding element to

the environment. When populating the table we use period of

CB oscillations of SET in order to find CSETQDG, CSET

PCG, and

CQDSET. In order to calculate CQD

SET, we apply potential to 2DEG

with zero potential at the QD gate. CSETR was calculated from

the charging energy of SET. Capacitances of the point con-

tact to the QD and SET are somewhat artificial, because the

point contact always has a good coupling to the 2DEG. We

found CSETPC � 21 aF using CB oscillations of the SET when

potential is applied to the 2DEG with the PC almost pinched

off. By solving the Laplace equation for the PC of

0.1� 0.1 lm2 size, we have got CSETPC � 30 aF, which is

close to the experimental value. The remaining capacitances

have been found from numerical solution of Laplace equa-

tion. One can do a consistency check of the capacitances by

calculating the effective capacitance between the SET and

the QD gate, when the QD is strongly coupled and isolated

from the reservoirs. The difference in effective capacitance

is seen as a change of period of the SET CB oscillations, Fig.

4(a). In the case of strong coupling, when CQDR !1, the

effective capacitance is equal to CSETQDG. Once the QD

becomes isolated from the reservoirs, the capacitance

becomes

Cef f � CSETQDG þ

CQDQDGCSET

R CQDSET

CQDR þ CQD

QDGCSETR

: (1)

From the experiment, we found Ceff¼ 2.9� 10�17 F. The

estimated value, �3.1� 10�17 F, is very close to the experi-

mental one.

IV. CHARGE SENSITIVITY OF QD-SET AND QD-PCDEVICES

The sensitivities of the SET and PC to charge fluctuation

at the QD are determined by the trans-conductance, dG/dVg,

and the capacitive coupling between the PC and the SET to

the QD, Vg being the voltage applied to the gate forming the

PC or the SET gate. One can combine two factors by introduc-

ing the sensitivity of the source-drain current to charge

variation at the QD, dI/dQ. Once the current noise in the

system, dI, is known, the detectable level of QD charge

excitation would be dQ ¼ dI=ð dIdQÞ. We found a moderate

sensitivity of the PC to charge excitation in region b of Fig. 2,

dI/dQ� 4� 106 A/C. Our set up has dI� 3 pA in a bandwidth

�1 kHz. The PC is then able to detect excitation of few elec-

trons, dQ� 4e, in/out of the QD. The sensitivity is enhanced

in a region where the CB oscillations are present. Typically,

the maxima of dI/dQ are higher by �15% in these regions.

The drawback is that the sensitivity becomes dependent on

the operation points: it is reduced at the extremes of the CB

oscillations and it is enhanced at the slopes. This makes the

photo-detector to be less stable in operation. The sensitivity of

the SET readout is only slightly higher, dI/dQ� 107 A/C. It is

constant along the boundary where the QD is formed. There is

�25% variation of sensitivity depending on the operation

point of SET itself. It follows from the analysis that the SET

should be able to detect excitation of dQ� e in a bandwidth

of 1 kHz. Indeed, the excitations of individual electrons are

clearly seen in our QD-SET device, they produce spikes close

to pinch-off region in Fig. 4(a).

V. CONCLUSIONS

In summary, we study the operation of the QD-SET-PC

device. We describe the system with the electrostatic model

TABLE I. Capacitances of the electrostatic circuit model of the device.

Symbol R denotes the total capacitance of a corresponding element to the

environment.

SET QD PC R QDG PCG

SET – 75 aF 21 aF 0.4 fF 2 aF 0.24 aF

QD 75 aF – 56 aF 0.25 fF 0.27 fF 24 aF

PC 21 aF 56 aF – 50 aF 50 aF 50 aF

FIG. 4. (a) CB oscillations of the SET current. Period of CB oscillations

changes from 81 to 5.5 mV when QD is decoupled from the reservoirs. (b)

Intensity plot of the CB oscillations in coordinates of VQD and VPC. The

dashed white line indicates the pinch-off boundary of the PC.

014322-4 Pelling et al. J. Appl. Phys. 112, 014322 (2012)

and determine a set of corresponding capacitances. The de-

vice is able to detect charge excitations of the QD with an ac-

curacy of less than one electron for the SET-QD and a few

electrons for PC-QD system. There are periodic oscillations

of the PC conductance close to pinch-off. We believe that

the oscillations are result of the Coulomb blockade of elec-

tron tunnelling in a quasi-1D channel, due to the formation

of a small dot of electrons, 35–100 nm size. The small dot

can be formed because of quantum well interface roughness.

ACKNOWLEDGMENTS

We acknowledge stimulating discussions with V. Gurto-

voi. This work is supported by the Solution Oriented

Research for Science and Technology (SORST) from the

Japan Science and Technology (JST), EPSRC Grants

EP/F005482/1, EP/G061432/1, and EU Integrated Project,

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